I recently made a hat out of hardware store twine, a project I’ll write about here. Soon, I hope. In creating the hat, I of course needed to know how large to make it. So I had Debra measure my head. I then divided that by π to get the diameter for the hat layout. Nothing magical there, but it got me to thinking about hat sizes.

In the US, hats are sized in 1/8 inch increments. Like 6-5/8 or 7-3/4. The number is computed by measuring the circumference of the head in inches 1 centimeter above the ears, and dividing by π. An alternate method is to take the circumference in centimeters and divide by 8.

What?

An example. Debra measured my head at 23 inches. Divided by π gives 7.32. That’s easy enough. Also, 23 inches works out to 58.42 centimeters. Divide that by 8 and you get … 7.30. Close enough for hat sizing! (So, yeah, my hat size is halfway between 7-1/4 and 7-3/8)

That works because to convert from inches to centimeters you multiply by 2.54. And (2.54 * π) is equal to 7.97. Or, showing my work . . .

Hat size = (circumference in inches) / π

Circumference in centimeters = (circumference in inches) * 2.54

Hat size = (circumference in centimeters) / (π * 2.54)

It’s kind of a cool estimating trick. You can estimate the diameter of any circle by dividing the circumference in centimeters by 8. The problem of course is the unit conversion: measure in centimeters and get the result in inches.

When I need to estimate with π, I use a two-step process. First, I multiply by 3. Then I add five percent. For example, a circle with diameter of 7 inches has a circumference (π * D) of 21.99 inches. Estimating, (7 * 3) = 21, and five percent of 21 is 1.05, giving me an estimated circumference of 22.05 inches. The mental arithmetic of *(times three plus five percent)* is a whole lot easier for me than *(times three point one four)*. Also, a simple *(times three)* is often enough to tell me what I need to know.

Or, if I’m dividing by π, I first divide by 3, which again is often enough to give me the answer I need. If I need a bit more precision I’ll subtract five percent. The result will be about one half percent less than the actual number. Again, close enough for a quick estimate.

I don’t often have to work with π^{2} when estimating, but a similar trick works pretty well. Just multiply by 10 and then subtract 10 percent. That is, π^{2} is about 9.87. So if you multiply by 10 and then subtract 10%, you’ll be about 3% high. Again, that’s close enough for a quick mental estimate.